The classic experiment to demonstrate this is the double-slit experiment. Take an opaque material, and cut in it two small slits. Shine on these slits some coherent light, such as that from a laser. On the side of the slits opposite the light source, place a light detector.
Firstly, you will observe diffraction and interference:
This demonstrates that light behaves like a wave. It does not behave like bullets -- if we fired bullets at these slits we would observe neither diffraction or interference.
You might notice that really we need just one slit to demonstrate that light is a wave: we will still see diffraction, which is sufficient to demonstrate wave-ness. However, the double slits also allows us to demonstrate that light is quantized, which is what is meant by the "particle" part of the duality. That is, light comes in discrete packets that can't be divided.
How do we know? Well, if we take this same apparatus, but make the laser very dim (by passing through absorptive materials, perhaps), and if we have a very sensitive light detector, we start to see discrete spots, as in (a):
We might make the laser so dim that only one photon strikes the detector per minute. Now here's the interesting thing: if you leave this experiment running for a long time, eventually the pattern recorded by the individual photons resembles the interference pattern from the wave above. This, despite that photons are traveling from the laser, to the detector one at a time!
Thus: light is a particle (in that it is quantized), and a wave (in that it can interfere). Always both: never just one.
Images from Wikipedia article: double-slit experiment